Abstract
We propose a novel approach of model selection for probability estimates that may be applied in time evolving setting. Specifically, we show that any discrepancy between different probability estimates opens a possibility to compare them by trading on a hypothetical betting market that trades probabilities.
We describe the mechanism of such a market, where agents maximize some utility function which determines the optimal trading volume for given odds. This procedure produces supply and demand functions, that determine the size of the bet as a function of a trading probability.
These functions are closed form for the choice of logarithmic and exponential utility functions. Having two probability estimates and the corresponding supply and demand functions, the trade matching these estimates happens at the intersection of the supply and demand functions.
We show that an agent using correct probabilities will realize a profit in expectation when trading against any other set of probabilities. The expected profit realized by the correct view of the market probabilities can be used as a measure of information in terms of statistical divergence.